What is the arc length of sqrt(2-x^2) – Calc II question?

A simple way is to imagine that the equation is an equatin of a circley = √(2 – x²) y² = 2 – x²x² + y² = 2with a center at the origin and radius of √2when x = 0, y = √2when x = 1, y = 1The angle formed with center as the angle between the two points is 45 degrees, or in other words, the arc length is 45/360 of the circumference of the circlearc length= 45/360 x 2 π r = (1/8)(2π)(√2)≈ 1.1107 units... Show More

Calc 2 Arc Length... Show More

(a.) ??a million+(dx/dy)^2 dy=arc length dx/dy=a million/2(36-y^2)^(-a million/2)(-2y)=-y/?36… Arc length=??a million+y^2/(36-y^2) dy from 0 to 3 (b.) ignore approximately my integration calculations for now for the reason which you haven’t any longer found out this way of integration yet. only be sure my answer including your graphing utility. ?6/?36-y^2 dy 6arcsin(y/6) evaluated at limits: 6arcsin(a million/2)-6arcsin(0)=6pi/6=pi... Show More

HiThe formula for arc-length is:L = (a to b) ∫ [√(1 + (dy/dx)^2)] dxa will be x = 0 and b will be x = 1. Since y = √(2 – x^2), we can find dy/dx to be:y = √(2 – x^2)dy/dx = (2 – x^2)’ * (1/2)(1/√(2 – x^2)) = -x/√(2 – x^2) Now, we just plug that into the formula. You’ll have to use a trig-substitution to solve this one.L = (0 to 1) ∫ [√(1 + (-x/√(2 – x^2))^2)] dx= (0 to 1) ∫ [√(1 + ((x^2)/(2 – x^2))] dx= (0 to 1) ∫ [√(((2 – x^2)/(2 – x^2)) + ((x^2)/(2 – x^2))] dx= (0 to 1) ∫ [√((2 – x^2 + x^2)/(2 – x^2))] dx= (0 to 1) ∫ [√(2/(2 – x^2))] dxLet x = √(2)sin(t), then dx = √(2)cos(t) dt and the limits change to 0 to π/4(0 to π/4) ∫ [√(2/(2 – (√(2)sin(t))^2))] √(2)cos(t) dt= (0 to π/4) ∫ [√(2/(2 – 2sin(t)^2))] √(2)cos(t) dt= (0 to π/4) ∫ [√(2/(2(1 – sin(t)^2)))] √(2)cos(t) dt= (0 to π/4) ∫ [√(1/(1 – sin(t)^2))] √(2)cos(t) dt= (0 to π/4) ∫ [√(1/cos(t)^2)] √(2)cos(t) dt= (0 to π/4) ∫ [√(sec(t)^2)] √(2)cos(t) dt= (0 to π/4) ∫ [sec(t)] √(2)cos(t) dt= (0 to π/4) ∫ [√(2)sec(t)cos(t)] dt= (0 to π/4) ∫ [√2] dt= [t√2] (0 to π/4)= (π/4)√(2) – (0)√(2)= (π/4)√(2) – 0= √(2)π/4≈ 1.111So the arc-length is about 1.111.I hope this helps!... Show More

(3 – x^2) dx3x – x^3 / 3 ](0,1) 3 – 1/38/3... Show More